Health Warning: (For Dummies) Please Do Not Eat Actual Radioactive Isotopes

Name______Class Period_____

Decay of Skittleium

Introduction:
The decay of radioactive isotopes creates interesting data. The isotope Skittleium is commonly found in vending machines throughout the world. The final decay elements of Skittleium are Skittles. We will be measuring the decay of Skittleium. Please dispose of all decay elements properly in a nearby oral cavity.

Health Warning: (For Dummies) Please do not eat actual radioactive isotopes.

Instructions: Obtain 1/4 Cup of Skittleium isotopes from the instructor. Count the isotopes. You need around 60 atoms of Skittleium to do this activity. Place isotopes in a box so that the “S” side is up. Cover box with a lid and shake vigorously. Take off lid, and remove all decayed Skittles (the ones that don’t have an “S” on top). Count how many Skittleium isotopes are still in box and record this number in data table below. Dispose of decay elements. Replace lid and shake again. Remove decayed Skittles, count and repeat until no Skittles remain in the box.

1) Enter the data in two lists in your graphing calculator. Perform linear regression and find the prediction equation and correlation.

Prediction Equation: ______
Correlation: ______

2) With this correlation, would you be willing to use your prediction equation to make predictions? Y N

3) Suppose you have actual data (like we do!). What is the first thing you should always do with that data before performing Linear Regression and making predictions? (Hint: See #4)

4) Graph the data and make an accurate sketch of the graph. Put the Round # on the x-axis and the Number of Isotopes on the y-axis. Describe the interesting features of the graph.

5) Add the Least Squares prediction line using a different color.

6) Should you use a line to make predictions for this data? Y N

7) Make a residual plot and make a small sketch of this plot. Are the residuals randomly scattered, or do you observe a pattern (circle one)? Random PatternReminder: Patterns are in Residual Plots are usually a BAD sign.

There are many, many equations that model curved data. Exponential models are often a good choice when trying to fit a curve to data, but often it is difficult to tell if the curve is really exponential. The shape of the decay curve of �S� up Skittles looks like an exponential curve, but our eyes are not very good at judging exactly what type of curve we are looking at. We need a good technique to check whether a curve is exponential. Our eyes are quite good at judging whether or not points lie along a straight line. So we will apply a mathematical transformation that changes exponential curves into lines�and curves that are not exponential will be transformed into something other than linear. We will be answering the question: �Is this curve exponential or not?�

8. If a variable grows exponentially, its logarithm grows ______(IPS p. 184).

9. Deviations from the overall pattern of exponential growth are most easily examined by ______

______(IPS p. 189).

10. Perform a logarithmic transformation on the following data, rounding to the nearest 10th:

ln {2.72, 7.39, 20.1, 54.6, 148.4, 403.4} = { ______, ______, ______, ______, ______, ______}

11. Try taking the logarithm of 0. What do you get? ______
To see why this is true, go to the Y= menu of your graphing calculator, and graph y= ln x. For x = 0, y = ______

12. Review the scatter plot of the Skittle data. Which set of data is the one that appears to be decaying
exponentially, the Round # or Number of Isotopes? (Duh!)
13. Perform a logarithmic transformation on the appropriate list of data. Example: ln (L3) STO �> L4.
If you get an error message, see #11 above. Sketch a graph the new scatter plot, using the transformed data on the Y-axis.

14. Find the regression equation for the transformed data, and sketch it on the graph.
r = ______
ln y^ = ______

15. Why should you write the new prediction line as
ln y^, and not just y^ ?

16. Can the transformed data be modeled by a linear equation? Y N
Be sure to check the residual plot!

17. Based on the transformed data, were your Skittles decaying exponentially? Y N
(HINT: Logarithms and exponents are inverse functions. One will "undo" the other, so if the logarithmic transformation straightened out the data, then it MUST have been exponential to start with.)

Retrieved online 11 August 2009 from Created August 2008. Revised for AP in August 2009.